Previous research by the PI and by others has succeeded in demonstrating both the existence and the nature of a long-run equilibrium stationary population when below-replacement fertility rates are assumed to be fixed in the face of a constant influx of immigrants. In particular, it is known that if, starting at some point, the age-specific birth and death rates of a population are held constant, if fertility is below replacement, and if the annual number of immigrants together with their age-sex composition are fixed, then that population will evolve in the long run to a stationary population whose size and age composition do not change. Our proposed new research is aimed at two related aspects of this problem: (1) to determine how long it takes for this long-run stationary population to become established under a variety of initial conditions and (2) in a comparative statics framework, to see how sensitive the long-run equilibrium population is to small changes in underlying assumptions about fertility, mortality and immigration. Standard tools of mathematical demography and partial differential calculus will be used to carry out the purposes of the research.